Dehgardi, N., Aram, H., Khodkar, A. (2018). The second geometric-arithmetic index for trees and unicyclic graphs. Iranian Journal of Mathematical Chemistry, 9(4), 279-287. doi: 10.22052/ijmc.2017.81079.1277

N. Dehgardi; H. Aram; A. Khodkar. "The second geometric-arithmetic index for trees and unicyclic graphs". Iranian Journal of Mathematical Chemistry, 9, 4, 2018, 279-287. doi: 10.22052/ijmc.2017.81079.1277

Dehgardi, N., Aram, H., Khodkar, A. (2018). 'The second geometric-arithmetic index for trees and unicyclic graphs', Iranian Journal of Mathematical Chemistry, 9(4), pp. 279-287. doi: 10.22052/ijmc.2017.81079.1277

Dehgardi, N., Aram, H., Khodkar, A. The second geometric-arithmetic index for trees and unicyclic graphs. Iranian Journal of Mathematical Chemistry, 2018; 9(4): 279-287. doi: 10.22052/ijmc.2017.81079.1277

The second geometric-arithmetic index for trees and unicyclic graphs

^{1}Department of Mathematics and Computer Science, Sirjan University of Technology, Sirjan, Iran

^{2}Department of Mathematics, Gareziaeddin Center, Khoy Branch, Islamic Azad University, Khoy, Iran

^{3}Department of Mathematics, University of West Georgia, Carrollton GA 30082

Abstract

Let $G$ be a finite and simple graph with edge set $E(G)$. The second geometric-arithmetic index is defined as $GA_2(G)=\sum_{uv\in E(G)}\frac{2\sqrt{n_un_v}}{n_u+n_v}$, where $n_u$ denotes the number of vertices in $G$ lying closer to $u$ than to $v$. In this paper we find a sharp upper bound for $GA_2(T)$, where $T$ is tree, in terms of the order and maximum degree of the tree. We also find a sharp upper bound for $GA_2(G)$, where $G$ is a unicyclic graph, in terms of the order, maximum degree and girth of $G$. In addition, we characterize the trees and unicyclic graphs which achieve the upper bounds.